| L(s) = 1 | − 3-s − 4·4-s − 2·9-s + 4·12-s − 12·13-s + 12·16-s − 2·19-s + 25-s + 5·27-s − 2·31-s + 8·36-s + 2·37-s + 12·39-s − 14·43-s − 12·48-s − 14·49-s + 48·52-s + 2·57-s − 24·61-s − 32·64-s − 4·67-s − 18·73-s − 75-s + 8·76-s − 20·79-s + 81-s + 2·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2·4-s − 2/3·9-s + 1.15·12-s − 3.32·13-s + 3·16-s − 0.458·19-s + 1/5·25-s + 0.962·27-s − 0.359·31-s + 4/3·36-s + 0.328·37-s + 1.92·39-s − 2.13·43-s − 1.73·48-s − 2·49-s + 6.65·52-s + 0.264·57-s − 3.07·61-s − 4·64-s − 0.488·67-s − 2.10·73-s − 0.115·75-s + 0.917·76-s − 2.25·79-s + 1/9·81-s + 0.207·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.556664276995089132763530544463, −8.232098142144935795769541453640, −7.62296727164466941005587284725, −7.31046477239498505884334187868, −6.60298576432161025427562908951, −5.88677489085139014953254885133, −5.44890211749766646861026772927, −4.92691899597547312483394760424, −4.59705496843248954156241201355, −4.39426797212802642244659436419, −3.14330660201336049402162930576, −2.95552129785071976456016297535, −1.69679652696223081739801604831, 0, 0,
1.69679652696223081739801604831, 2.95552129785071976456016297535, 3.14330660201336049402162930576, 4.39426797212802642244659436419, 4.59705496843248954156241201355, 4.92691899597547312483394760424, 5.44890211749766646861026772927, 5.88677489085139014953254885133, 6.60298576432161025427562908951, 7.31046477239498505884334187868, 7.62296727164466941005587284725, 8.232098142144935795769541453640, 8.556664276995089132763530544463
